Vaught’s Conjecture on the number of countable models

Determine whether, for every countable first-order theory in a countable signature, the set of isomorphism classes of its countable models is either at most countable or of cardinality 2^{aleph_0} (Vaught’s Conjecture).

Background

The paper opens by situating its contributions within classical model theory, highlighting Vaught’s Conjecture, which proposes a dichotomy for the number of non-isomorphic countable models of a countable first-order theory: either countable or continuum many.

Morley’s theorem provides a weaker dichotomy (at most aleph_1 or continuum many), motivating the focus on absolute formulations and second-order generalizations addressed in the paper.